The power of choice in network growth

The "power of choice" has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of tree and network growth. In our models each new node has k randomly chosen contacts, where k > 1 is a constant. It then attaches to whichever one of these contacts is most desirable in some sense, such as its distance from the root or its degree. Even when the new node has just two choices, i.e., when k=2, the resulting network can be very different from a random graph or tree. For instance, if the new node attaches to the contact which is closest to the root of the tree, the distribution of depths changes from Poisson to a traveling wave solution. If the new node attaches to the contact with the smallest degree, the degree distribution is closer to uniform than in a random graph, so that with high probability there are no nodes in the network with degree greater than O(log log N). Finally, if the new node attaches to the contact with the largest degree, we find that the degree distribution is a power law with exponent -1 up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case, we need k >> 1 to see a power law over a wide range of degrees.Comment: 9 pages, 4 figure

Ballistic Coalescence Model

We study statistical properties of a one dimensional infinite system of coalescing particles. Each particle moves with constant velocity $\pm v$ towards its closest neighbor and merges with it upon collision. We propose a mean-field theory that confirms a $t^{-1}$ concentration decay obtained in simulations and provides qualitative description for the densities of growing, constant, and shrinking inter-particle gaps.Comment: 4 pages, 2 column Revtex, 5 figures include